Spectral gaps and exponential integrability of hitting times for linear diffusions
Let \((X_t) _ {t \geq 0}\) be a regular, positively recurrent and conservative Markov process with state space \(\mathbb{R}\). Using the quantities \[ B_ {a}^ {+} : = \sup _ {x \geq a} m ( (x, +\infty ))(S(x) - S(a)) ;\;B_ {a}^ {-} : = \sup _ {x \leq a} m ( (-\infty , x))(S(a) - S(x)), \] where \(S\) is a scale function of \(X\) and \(m\) is the speed measure associated with \(S\), the authors are able to provide a two-sided estimate for the exponential moments of hitting times. Then, they establish a direct relation between exponential moments and spectral gaps of generators associated with \(X\), improving the classical bounds. The final part of the paper illustrates the authors' results for some particular processes, such as drifted Brownian motions and Ornstein-Uhlenbeck processes.
scientific article; zbMATH DE number 5956523
scientific article